| L(s) = 1 | − 0.593·2-s + 3-s − 1.64·4-s + 3.15·5-s − 0.593·6-s + 2.16·8-s + 9-s − 1.87·10-s − 6.54·11-s − 1.64·12-s + 0.692·13-s + 3.15·15-s + 2.01·16-s − 6.84·17-s − 0.593·18-s + 2.91·19-s − 5.19·20-s + 3.88·22-s − 23-s + 2.16·24-s + 4.93·25-s − 0.411·26-s + 27-s + 2.94·29-s − 1.87·30-s + 0.539·31-s − 5.52·32-s + ⋯ |
| L(s) = 1 | − 0.419·2-s + 0.577·3-s − 0.823·4-s + 1.40·5-s − 0.242·6-s + 0.765·8-s + 0.333·9-s − 0.591·10-s − 1.97·11-s − 0.475·12-s + 0.192·13-s + 0.813·15-s + 0.502·16-s − 1.65·17-s − 0.139·18-s + 0.669·19-s − 1.16·20-s + 0.828·22-s − 0.208·23-s + 0.441·24-s + 0.986·25-s − 0.0806·26-s + 0.192·27-s + 0.547·29-s − 0.341·30-s + 0.0968·31-s − 0.976·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.593T + 2T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 13 | \( 1 - 0.692T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 - 0.539T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + 9.26T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 5.30T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 + 0.587T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 9.77T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 - 8.14T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.483744312764110504071867593737, −7.71031098478768384355910467659, −6.85865796778751931390668926931, −5.88461499571492251115469032952, −5.09340951753662146895236828680, −4.61911668635515404236791900942, −3.28260124237495552259616098795, −2.40326630875390028422436825025, −1.59749037861071601519040793623, 0,
1.59749037861071601519040793623, 2.40326630875390028422436825025, 3.28260124237495552259616098795, 4.61911668635515404236791900942, 5.09340951753662146895236828680, 5.88461499571492251115469032952, 6.85865796778751931390668926931, 7.71031098478768384355910467659, 8.483744312764110504071867593737
